Within this approach, the fundamental kinematical variables are the spinors,
which can be considered as the building blocks for the amplitudes of massless particles [1]}.
The starting point is to express the 4-momenta, \(p_{\mu }\) , as a \(2\times 2\)  matrix:
\( p_{\alpha \dot{\alpha }} = \sigma ^{\mu }_ {\alpha \dot{\alpha }} p_{\mu }\)  (\(\alpha =1,2\)  and \(\dot{\alpha }=\dot{1},\dot{2}\) ),
using the using the Pauli matrices \(\sigma ^{\mu }\) .
For massless particles \(det(p)=0\) , and then the momentum matrix can be factorized in terms of Weyl spinors, as:
\( p_{\alpha \dot{\alpha }} = \chi _{\alpha } \tilde{\chi }_{\dot{\alpha }}\) . It can be convenient to use Dirac notation,
with \(\chi _{\alpha } (p_i) \rightarrow | i \rangle \)  and \(\tilde{\chi }_{\dot{\alpha }} (p_i) \rightarrow [ i | \) , both for the momentum and for the
wave functions associated with external fermions. The inner products of spinors are defined as:
\(\langle i j \rangle = \chi ^\alpha \chi _\alpha \)  and \( [i j ] = \tilde{\chi }_{\dot{\alpha }} \tilde{\chi }^{\dot{\alpha }}\) , such that:
\(\langle j i \rangle [i j ] = 2 p_i \cdot p_j\) .
